To use a wand, you either need to have the spell from the wand on your Class list, or have a very high Use Magical Device skill check.
If you want to use wands and wear armor, from the Core classes, than by all means, be a Bard because they can wear light armor with no problems. Once you get to Medium and heavy, you start to have some problems, but not all problems.
Arcane Spell Failure only occurs with spells with somatic components. This means that every spell without a somatic component can be cast by any arcane caster class in full plate, with no spell related consequences.
Arcane Spell Failure had nothing to do with using a wand. Using a wand is not casting a spell. So, wear full plate as whatever class you want and use a wand, as long as you can trigger the wand.
Now the
Battle Caster Feat will permit you to wear armor of one category higher than you can normally wear, without incurring arcane spell failure. The prereq. is that you can cast in some form of armor, which a Bard can do. Take this feat once to use Medium, take this feat twice to wear everything. It doesn't allow you to use shields, but it'll give you the full plate.
There is no need to multiclass to accomplish the things in this post.